Adding an image to a mirrored object
You could use a UV Project Modifier, to project a texture on to your mirrored model. 

If you are using the UV map (without the Project Modifier) then no there is no way. You will have to apply the mirror modifier.
With a mirror modifier on a model, there is only half the mesh, and thus half the UVs to texture with. So anything that is one the first half, will be mirrored to the other half, including materials and textures.